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In this paper we address the problem of unsupervised gaze correction in the wild, presenting a solution that works without the need for precise annotations of the gaze angle and the head pose. We have created a new dataset called CelebAGaze, which consists of two domains X, Y, where the eyes are either staring at the camera or somewhere else. Our method consists of three novel modules: the Gaze Correction module (GCM), the Gaze Animation module (GAM), and the Pretrained Autoencoder module (PAM). Specifically, GCM and GAM separately train a dual in-painting network using data from the domain $X$ for gaze correction and data from the domain $Y$ for gaze animation. Additionally, a Synthesis-As-Training method is proposed when training GAM to encourage the features encoded from the eye region to be correlated with the angle information, resulting in a gaze animation which can be achieved by interpolation in the latent space. To further preserve the identity information~(e.g., eye shape, iris color), we propose the PAM with an Autoencoder, which is based on Self-Supervised mirror learning where the bottleneck features are angle-invariant and which works as an extra input to the dual in-painting models. Extensive experiments validate the effectiveness of the proposed method for gaze correction and gaze animation in the wild and demonstrate the superiority of our approach in producing more compelling results than state-of-the-art baselines. Our code, the pretrained models and the supplementary material are available at: https://github.com/zhangqianhui/GazeAnimation.